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Generalized Estimation Methods for Non-i.i.d.
Binary Data: An Application to Dichotomous
Choice Contingent Valuation
SOO-IL KIM
[email protected]
Dept. of Energy Policy Studies, Korea Energy Economics Institute
665-1 Naeson 2-dong, Euiwang-si, Kyonggi-do, 437-713, Korea
Timothy C. Haab
[email protected]
AED Economics, The Ohio State University
2120 Fyffe Road, Columbus, Ohio United States 43210
Selected Paper prepared for presentation at the American Agricultural Economics
Association Annual Meeting, Providence, Rhode Island, July 24-27, 2005
Copyright 2005 by SOO-IL KIM and Timothy C. Haab. All rights reserved. Readers may
make verbatim copies of this document for non-commercial purposes by any means,
provided that this copyright notice appears on all such copies.
Generalized Estimation Methods for Non-i.i.d.
Binary Data: An Application to Dichotomous
Choice Contingent Valuation
SOO-IL KIM and Timothy C. Haab
Abstract: We challenge the assumption of i.i.d random utility across alternatives
embedded in typical applications of logit models to dichotomous choice contingent
valuation data. Using a Gumbel mixed distribution which nests a number of traditional
models, we show that the logistic distribution is not a suitable distribution for contingent
valuation analysis.
1
1. Introduction
Including dichotomous choice contingent valuation (DCCV), widely different fields
of economics have used the binary data based on the random utility model. The binary
choice in DCCV is “yes (or one)” if the random utility after environmental change is still
greater than that of the current state, and “no (zero)” otherwise. With the assumption of i.i.d.
type I extreme value for the distribution of the unobserved term, the random utility models
can be estimated through a simple logit model (See Haab and McConnell 2002).
The simplicity and robustness of the estimation model, however, are the result of
strong assumptions or constraints on the decision model rather than the natural outcome of
correct specification of the model1. The main problem is that i.i.d. assumption across
alternatives can be far from the real choice situation. First, since the state after
environmental change is uncertain to the respondent in spite of surely increasing
environmental quality, the variance of the additive error term in the proposed state may be
different from that in the current random utility. Second, the existence of alternative project
that respondents prefer but the researcher does not consider in the CV survey, can lead the
respondent to refuse the proposed project even though respondent agrees with the change in
environmental quality. In this situation, the simple logit is not suitable estimation model
and yields an incorrect measure of parameters or welfare change.
Undoubtedly, there has been a series of studies to relax the i.i.d. assumption in the
logit model. For example, the heteroskedastic extreme value model has been suggested in
the transportation (Bhat 1995) and marketing literatures (Allenby and Ginter 1995) to
incorporate heteroskedasticity across alternatives into the multinomial or conditional logit
models. However, no literature has paid attention to the strict assumption of identical error
distributions across alternatives in the choice set. Since alternatives are only two and the
scale and level of the utility is immaterial, variance-covariance parameters cannot be
estimated with the binary choice by conventional models including generalized extreme
values such as nested logit, paired combinatorial model, etc2.
1
We focus only on the logit model. When we assume the bivariate normal distribution, the choice probability
still follows the univariate normal. The probit model, however, can identify only one parameter related with
error distribution because of normalization. For details, see Train (2003).
2
Note that these models have only J(J – 1)/2 – 1 covariance parameters after normalization, where J is the
number of total alternatives
2
In this paper, we relax the identical and independent disturbance assumption of the
random utility model by utilizing Gumbel mixed model. Gumbel mixed model is an
asymptotic bivariate distribution of maxima, which provides us a scale parameter of one
state normalized by the scale parameter of the other state and an association parameter
related with correlation coefficient. The derivation of Gumbel mixed model is explained in
the section 2. In section 3, we introduce two estimation techniques with Gumbel mixed
model: approximation using Gaussian quadrature and simulation using mixed logit model.
The generalized estimation procedures are applied to two contingent valuation studies in
the section 4. Section 5 applies the generalized estimation method into the alternative
choice model (willingness to pay function from expenditures difference), in which we
found the same result as the random utility model. The section 6 concludes the analysis.
2. Gumbel Mixed Model of Bivariate Extreme Values Distribution
Including Gumbel (1960, 1961), Gumbel and Mustafi (1967) and Tiago de Oliveira
(1980, 1983), a series of papers has introduced several bivariate extreme value distributions
including the Gumbel mixed model which is one of differentiable bivariate extreme value
distributions3. Let F ( ε 0 , ε1 ) be a asymptotic distribution of bivariate extreme values of
maxima for ε 0 and ε1 with Gumbel margins, F ( z ) . The asymptotic distribution of
bivariate maxima is defined as
(1)
F ( ε 0 , ε1 ) = ⎡⎣ F ( ε 0 ) F ( ε1 ) ⎤⎦
k (τ )
where k ( ⋅) is called the dependence function representing the asymptotic connection
between ε 0 and ε1 , and τ is reduced difference defined as ε 0 / θ 0 − ε1 / θ1 . θi is a scale
factor and the location factor is assumed to be equal to zero.
Different bivariate distributions are derived using different dependence functions, of
which the Gumbel mixed model has k (τ | λ ) = 1 − λ exp (τ ) / (1 + exp (τ ) ) , where λ is an
2
3
For other examples of parametric families of bivariate extreme value distributions, see Kotz and Nadarajah,
2000. Applications of Gumbel mixed model can be found in the hydrological engineering studies (Yue 2000,
Yue et al. 1999).
3
association parameter4. Plugging the dependence function into equation (1) and using the
definition of Gumbel margins, Fε i ( z ) = exp ( − exp ( − z / θi ) ) , the Gumbel mixed model is
expressed as
(2)
⎡ ⎛
⎤
⎛ ε ⎞
⎛ ε ⎞⎞
λ
F ( ε 0 , ε1 | Γ ) = exp ⎢ − ⎜ exp ⎜ − 0 ⎟ + exp ⎜ − 1 ⎟ ⎟ +
⎥
⎜
⎟
⎢⎣ ⎝
⎝ θ1 ⎠ ⎠ exp ( ε 0 / θ 0 ) + exp ( ε1 / θ1 ) ⎦⎥
⎝ θ0 ⎠
where Γ is a parameter set of scale factor ( θ 0 , θ1 ) and association factor (λ). Note that the
expected value and the variance of independent extreme value ε i are E ( ε i ) ≈ 0.57722θi
and Var ( ε i ) = θi2π 2 / 6 . Figure 1 shows the contour of the Gumbel mixed bivariate
distribution function with λ = 0.5. For λ = 0 , the joint distribution is independent such that
F ( ε 0 , ε1 ) = F ( ε 0 ) F ( ε1 ) . The correlation coefficient is a function of the association
parameter λ.
From the Gumbel mixed distribution, several important distributions are derived;
probability density function, conditional distribution and distribution of reduced difference.
The probability density function is derived by differentiating (2) with respect to ε 0 and ε1
such that
f ( x, y )
(3)
⎡
⎧
F ( x, y ) ⎢ 2λ e x / θ1 + y / θ2
λ e x /θ1
⎪ − x / θ1
e
=
+
−
⎨
θ1θ 2 ⎢ e x / θ1 + e y / θ2 3 ⎪
e x / θ1 + e y / θ2
⎩
⎣⎢
(
)
(
)
⎫⎧
λ e y /θ2
⎪ ⎪ − y / θ2
e
−
2 ⎬⎨
e x /θ1 + e y /θ2
⎭⎪ ⎩⎪
(
)
⎫⎤ .
⎪⎥
2⎬
⎥
⎭⎪⎦⎥
The contour of probability function is shown in Figure 2. As can be seen in Figure 1 and 2,
the bivariate extreme value distribution is upper-right skewed.
[Figure 1-2 located here]
The conditional cumulative distribution function of the Gumbel mixed model is
(4)
⎧
exp ⎡⎣ 2ε1 / θ1 + exp ( −ε1 / θ1 ) ⎤⎦ ⎫⎪
⎪
Fε 0 |ε1 ( ε 0 ) = F ( ε 0 , ε1 ) ⎨exp ⎡⎣exp ( −ε1 / θ1 ) ⎤⎦ − λ
2⎬
⎡⎣exp ( ε 0 / θ 0 ) + exp ( ε1 / θ1 ) ⎤⎦ ⎪⎭
⎪⎩
4
The logistic model, one of differentiable bivariate extreme value distribution, is derived using the difference
function of k (τ | λ ) = ⎡1 + exp ( −τ / (1 − λ ) ) ⎤1− λ / ⎡1 + exp ( −τ ) ⎤ . Unfortunately, the logit model, i.e. the generalized
⎣
⎦
⎣
⎦
extreme value with two alternatives, cannot identify the association factor λ.
4
from fε 0 |ε1 = f ( ε 0 , ε1 ) / fε1 ( ε1 ) (Yue 2000). The distribution function of reduced difference
is derived as (Tiago de Oliveira 1980)
(5)
(1 + exp (τ ) ) − λ .
D (τ | λ ) =
1 + exp (τ ) (1 + exp (τ ) ) − λ exp (τ )
2
exp (τ )
2
Figure 3 and 4 show the cumulative distribution and probability function of the reduced
difference with various λ. The probability density function of reduced difference is
symmetric around zero mean. For λ = 0 , i.e. independent case, the conditional distribution
(4) reduces to be a univariate type I extreme value and the difference distribution (5)
becomes a logistic distribution.
[Figure 3-4 located here]
3. Random Utility and the Probability of Binary Choice
3.1. Random Utility Model
A standard random utility consists of two parts; a systematic component observable
to researcher and an error component that is known to respondent but not necessarily. Let
the random utility of individual n with alternative i be U in = Vi ( I n , zn ) + ε in , where Vi is the
systematic component and ε i is the error component. Alternatives in the binary choice set
are the proposed state representing to accept the policy and the current state without change
indicating to reject the policy5. The systematic part is the function of respondent’s income
( I n ) and the vector of respondent’s characteristics and choice attributes ( zn ). The
probability of choosing the state one is;
(6)
P1n = P (U 0 n < U1n ) = P (V0 n + ε 0 n < V1n + ε1n ) = P ( ε 0 n < vn + ε1n )
where vn = V1n − V0 n . Further progress in estimation is feasible by specifying a parametric
form for both of the systematic component and the error distribution in equation (6). The
systematic component is usually assumed linear in parameters even though only linearity in
income is sufficient. The error component in the standard additive random utility for
discrete choice case is assumed independent and identical distribution over states
5
We assumes that ‘to be uncertain’ responses are grouped as ‘no’ response for conservative reason. For
details of ‘uncertain’ response issue, see Carson et al. 1998; Groothuis and Whitehead 1998.
5
(Karlström 1999). With i.i.d. type I extreme value (or Gumbel) distribution, the derivation
of the logistic distribution for the difference of two identical extreme values is
straightforward. In addition, McFadden (1974) also shows that the logit formula for the
choice probabilities implies extreme value distribution for the random utility.
The classical assumption about the additive error components, however, may be
wrong because of several reasons. For example, uncertainty in the future, reliability on the
implementation and result of the project, etc, can be possible source that respondent accept
the proposed and current states differently. More uncertainty in the proposed state
introduces larger variance of the distribution. The proposed state is random and
unobservable even to the respondent, which is different from the unobservability of the
current state. Another possibility of violation of the assumption is that, if respondents have
alternative options instead of the proposed policy that may be unknown to researcher, the
response of reject represents either staying without change (the current state) or changing
through other process (the future state possibly for different level of environmental
quality)6. In addition to the possible heteroskedasticity due to unknown alternatives, the
current and proposed states may be correlated if the unknown alternatives have the same
goal of the environmental change with the proposed policy in the survey. Because of those
reasons, we name the current state as the reference state to avoid misinterpretation.
3.2. Approximation of the Log Likelihood
Regardless of the heteroskedasticity and correlation, the choice probability in the
equation (6) can be expressed as an integration of the conditional distribution over marginal
distribution;
(7)
P1n = ∫
ε1 =+∞
ε1 =−∞
Fε 0 |ε1 ( vn + ε1 ) f ( ε1 ) d ε1 .
6
Train (2003) defined three characteristics that alternatives in the choice set should satisfy: exclusiveness,
exhaustiveness and countable finiteness. To vote for and vote against are mutually exclusive and finite. For
exhaustiveness, the current state without change includes not only the state without change but also all
possible changes except the policy proposed in the survey. Furthermore, NOAA panel report (Arrow et al.
1993) recommends the reminder of substitute commodities among guideline for designing contingent
valuation questions, such as other comparable natural resources or the future state of the same resource to
assure that respondents have the alternatives clearly in mind (Haab and McConnell, 2002). Haab and Hicks
(1999) has broadly surveyed the choice set issues in recreation demand modeling.
6
The equation (7) is a general expression of the choice probability nesting a simple logit,
heteroskedastic extreme values and common scale factor model. Since the general
expression of bivariate extreme value model in equation (7) does not have the closed form
for the integration, we need to approximate or simulate the choice probability.
As already shown in the multinomial heteroskedastic case by Bhat (1995), the
approximation procedure utilizing Gaussian quadrature can provide a fast and highly
accurate proxy of the choice probability7. Define a new variable such that
u = exp ( − exp ( − w ) ) , thus w = − ln ( − ln u ) and du = exp ( − w ) exp ( − exp ( − w ) ) dw 8. The
new variable u is the form of cumulative distribution of extreme value and has the support
of [ 0,1] . This transformation enables us to approximate the choice probability much easier
through Gaussian-Legendre quadrature. Let ε1 = θ1w1 and γ = θ1 / θ 0 , then the conditional
density and marginal probability functions are Fε 0 |ε1 ( vn + ε1 ) = Fε 0 |ε1 ( vn + γ w1 ) and
f ( ε1 ) = f ( w1 ) / θ1 . The arguments in the conditional probability is normalized by θ 0 .
Plugging the new variable u into the choice probability function, the choice probability
becomes P1n = ∫
u =1
u =0
G ( vn , u ) du since d ε1 = θ1dw1 , where G ( vn , u ) = Fε 0 |ε1 ( vn − γ ln ( − ln u ) ) .
The integration can be substituted by Gaussian-Legendre quadrature such as
∫
u =1
u =0
G ( vn , u ) du ≈ ∑ l =1 ξl G ( vn , ul ) = Pˆ1n
L
where ξl and ul are L weights and support points (abscissas) of Gaussian-Legendre
quadrature. The log likelihood function is approximated as
N
{
}
log L = ∑ yn log ∑ l =1 ξl G ( vn , ul ) + (1 − yn ) log 1 − ∑ l =1 ξl G ( vn , ul ) .
L
n =1
L
3.3. Simulation of the Log Likelihood
The mixed logit model introduced into recreation model by Train (1998, 1999)
directly simulates the choice probability in the equation (6). Let the true random utility to
7
Alternatively, Allenby and Ginter (1995) also suggest the Bayesian estimation procedure for heteroskedastic
extreme values.
8
Bhat (1995) uses the transformation of u = exp ( − w) with the support of [0, ∞ ] and applies a GaussianLaguerre quadrature.
7
be U in = ς in′ zin , where zin′ = ( xin′ , d i ) and ς in′ = ( β in′ , ε in ) . By rescaling the utility upward
sufficiently (s) and adding an i.i.d. extreme value terms on both sides, the resulting choice
probability is expressed such as
⎛
⎞
exp ⎡⎣(ς 1′n / s ) z1n ⎤⎦
⎟ f (ς ) d ς
P1n = ∫ ⎜
⎜∑
⎡(ς ′jn / s ) z jn ⎤ ⎟
exp
⎣
⎦⎠
⎝ j =0,1
where f (ς ) is a joint density of β jn and ε jn 9. Suppose that coefficients of systematic part
of utility are invariant across individual ( β in = βi ) and the joint density of f (ς ) is a
bivariate distribution of ε 0 n and ε1n . Then the mixed logit model becomes
P1n = ∫ L1n (ς ) f ( ε 0 , ε1 ) d ε 0 , ε1
(8)
where
L1n (ς ) =
(9)
exp ⎡⎣vn / s + ( ε1n − ε 0 n ) / s ⎤⎦
1 + exp ⎡⎣ vn / s + ( ε1n − ε 0 n ) / s ⎤⎦
.
Because of the equivalence to the logit smoothed-AR simulator (McFadden 1989),
the estimation of the mixed model follows the simulation procedure of ‘logit kernel probit’
(Ben-Akiva and Bolduc 1996) adjusted simply for the bivariate extreme values. However,
since the random draw from a bivariate extreme value distribution is unavailable, we
employ an importance sampling procedure with Halton sequence to simulate the random
draw from bivariate extreme values. The importance sampling provides simulated random
variables with correlation and heteroskedasticity by transforming the original density,
named target density, into a density from which it is easy to draw, named a proposal density
(Train 2003).
Let g ( ε i ) be a univariate extreme value distribution. Define the weight as
9
The mixed logit model, usually, has employed a joint distribution of parameters β in the systematic
component of random utility. The probability function of the random parameter is defined to be
⎛ exp ( β ′ x ) ⎞
in
⎟φ (β )d β
Pin = ∫ ⎜
⎜
⎝
∑
j
exp ( β ′ x jn ) ⎟
⎠
where φ ( ⋅ ) is the distribution function of parameters which can be flexibly assumed such as a normal
(Provencher and Bishop 2004), lognormal (Bhat 2000), uniform or triangular (Train 2001) distribution. The
mixed logit model can be applied to any choice model with any degree of accuracy by assuming appropriate
distribution (Train 2003, McFadden and Train 2000). By assuming that parameters have an individual and
alternative specific randomness, the mixed model relaxes the IIA assumption and represents any pattern of
substitution among alternatives.
8
(10)
f ( ε 0 , ε1 )
g ( ε 0 ) g ( ε1 )
⎧⎪
⎫⎪
λ
= Ψ ( ε 0 , ε1 ) exp ⎨( ε 0 / θ 0 + ε1 / θ1 ) +
⎬
exp ( ε 0 / θ 0 ) + exp ( ε1 / θ1 ) ⎭⎪
⎩⎪
where f ( ε 0 , ε1 ) is the joint density in the mixed logit model and
Ψ ( x, y ) =
2λ e x / θ0 + y / θ1
(e
x / θ0
+ e y / θ1
)
3
⎧
λ e x / θ0
⎪
+ ⎨e − x / θ 0 −
e x / θ0 + e y / θ1
⎩⎪
(
)
⎫⎧
λ e y / θ1
⎪ ⎪ − y / θ1
−
e
⎬
⎨
2
e x / θ0 + e y / θ1
⎭⎪ ⎩⎪
(
)
⎫
⎪
.
2 ⎬
⎪⎭
Using the fact that ε i = θi wi and g ( ε i ) = (1/ θ i ) g ( wi ) , the choice probability of mixed logit
model becomes
⎧⎪
⎫⎪
λ
Pˆ1n = ∫ L1n (ς ) Ψ ( w0 , w1 ) exp ⎨( w0 + w1 ) +
⎬ d w0 , w1 ,
exp ( w0 ) + exp ( w1 ) ⎭⎪
⎩⎪
since multiplying the integrand of equation (8) by g ( ε ) / g ( ε ) does not change the original
choice probability.
Application of importance sampling to the mixed logit model is as follows: (1) Take
draws for w0 and w1 from a standard extreme value distribution and construct twodimensional independent random variables. In this first step and through the repetition,
Halton sequence is used to draw standard extreme values10. (2) For this draw, calculate the
logit formula, L1n with prespecified scaling factor (s), and the weight function of the
equation (10). (3) Repeat two steps enough times and take the average of the result,
1
Pˆ1n = ∑ r P1n , which is an unbiased estimate of the choice probability with correlation
R
and heteroskedasticity. The probability of choosing the alternative zero is Pˆ0 n = 1 − P̂1n . The
simulated log likelihood function becomes
(
N
)
log L = ∑ yn log Pˆ1n + (1 − yn ) log 1 − Pˆ1n .
n =1
3.4. Estimation of Welfare Change
We define the expected welfare change (willingness to pay for the environmental
change) as the expected maximum income that equates the expected random utility in two
10
Halton sequence reduces the number of draws and the simulation error associated with a given number of
draws. The simulation error with 125 Halton draws is smaller than even with 2000 random draws. See Bhat
(1999), Train (1999, 2003) and Greene (2002).
9
states11. Assume that the systematic component of the random utility is linear in the income
and the marginal utility of income is constant (α) across individuals and states, i.e. no
income effect, then the expected willingness to pay becomes
E (WTPn ) =
(11)
1
α
vn +
1
α
E ( ε1n − ε 0 n ) .
Due to the linearity assumption of the income, the income variable is not included in vn .
While the expectation of the error term of the logit model is zero by including a constant
term in the systematic component, the expected value of error terms in equation (11) is not
zero. As explained below, it is much convenient for estimating equation (11) to remain the
expectation term.
In general case, we can estimate the expectation of error differences through a
simulation procedure using estimated relative scale and association parameters since the
exact moment of error difference in Gumbel mixed model is unknown. Note that the
expected value of error difference is the integration of random variables over Gumbel
mixed bivariate probability;
E ( γ w1n − w0 n ) = ∫ ( γ w1n − w0 n ) f ( w1n , w0 n ) d w1n , w0 n .
By reapplying importance sampling procedure with Halton sequence, the expected
willingness to pay becomes
E (WTPn ) = xn′
β 1 ˆ
+ E ( γ w1n − w0 n ) .
α α
since E ( ε1n − ε 0 n ) / α is equivalently E ( γ w1n − w0 n ) / α .
Except the general case, however, the expected willingness to pay can be exactly
calculated. In the heteroskedastic case, the expectation of ε1n − ε 0n is approximately
0.57722 ⋅ (θ1 − θ 0 ) , providing the final expression of the expected willingness to pay as
E (WTPn ) ≈ vn / α + 0.57722 ( γ − 1) / α . For identical cases, the willingness to pay is simply
E (WTPn ) = vn / α .
11
A series of papers has investigated the correct welfare measure consistent with the microeconomic theory.
The welfare measurement is incorrect if we estimate the models using the incorrect choice set (Kaoru et al.
1995). More seriously, a large difference of amount of money in a cost-benefit analysis has been found even
though the welfare estimates from different model are similar (Hau 1986, Herriges and Kling 1999, Karlström
1999).
10
4. Applications to Dichotomous Choice Contingent Valuation Study
To test classical assumption, we apply Gumbel mixed model to the previous
dichotomous choice CV studies. The study used in the estimation includes the sewage
treatment in Barbados and the wastewater disposal system in Montevideo, Uruguay
(McConnell and Ducci, 1989)12. Observations in data were 1276 for Montevideo and 426
for Barbados data. For mixed logit model, the rescaling factor s was set to be 0.3 and the
simulation was iterated 125 times. The association parameter (λ) was constrained to be
between zero and one, and the relative scale factor (γ) was restricted to be nonnegative in
the CML procedure of Gauss program13.
Table 1 and Table 2 show the estimation results of random utility model with
Barbados and Montevideo data, respectively. The results consist of three sets; simple logit
model in the second column, the result of bivariate extreme values in the third to sixth
column and the result of mixed logit in the last four columns.
[Table 1 located here]
In Table 1, the first part of results of each estimation method is for the constrained model
with independent and identical error ( γ = 1 , λ = 0 ) that is theoretically equivalent to the
simple logit model. The constrained bivariate extreme value model with γ = 1 and λ = 0
provides exactly same parameter estimates with the simple logit model while the
constrained mixed logit model with γ = 1 and λ = 0 has slightly different estimates from
the simple logit model. However, both estimation models fail to estimate parameters with
the constraint of λ = 0 due to too large relative scale estimate. When the correlation is
allowed in estimation, i.e. in general model and constrained model with γ = 1 , the
association parameter is different from zero but not statistically significant. In addition, the
relative scale parameter is not statistically different from one. LR statistics fail to reject the
constraints for homoskedasticity or independence. Barbados data shows that the assumption
of independent and identical distribution is suitable for estimation of random utility.
[Table 2 located here]
12
The data is available in Haab and McConnell (2002).
The positive constraint can be assigned in the model by transforming the parameter such as exponential
term. However, the estimation results for other parameters are not different and zero estimate of relative scale
implies extreme difference of scale terms.
13
11
Table 2 also shows that bivariate extreme value model with constraints of γ = 1 and
λ = 0 provides the estimation result closer to logit model than mixed logit model does. In
both of bivariate extreme value and mixed logit models, association parameter estimates are
statistically indifferent from zero except one case of mixed logit model. The relative scale
estimates, however, are zero implying that the variance of the reference state is extremely
larger than that of the proposed state. The association parameter estimates are not
statistically different from zero. LR statistics fails to reject the constraint of independence
( λ = 0 ), but heteroskedasticity is statistically significant.
[Table 3 located here]
Table 3 shows the sample average of the expected willingness to pay from Table 1
14
and 2 . RD indicates the random utility model and ED represents the expenditure
difference model result explained in the next section. In spite of similar parameter estimates
among different constrained models, the welfare measure from the change of environmental
quality varies enormously depending on the relation of error terms. For instance, the sample
average of the expected willingness to pay in Montevideo is estimated around -28 ~ -26
when γ = 1 is imposed, but it is estimated -81 ~ -65 without the constraint of γ = 1 .
Unfortunately, due to the failure of estimation, the willingness to pay cannot be estimated
in two heteroskedastic models of random utility with Barbados data. However, the expected
willingness to pay with the homoskedasticity constraint is also similar to logit model since
independence has been found in most cases.
5. Expenditure Difference Model
An alternative model of explaining respondent’s choice in dichotomous choice
contingent valuation is the willingness to pay function derived from the expenditure
(
)
functions. Let the minimum expenditure of individual n be m0 n = m q 0 , un0 at the reference
state and m1n = m ( q1 , un0 ) at the proposed state where q i is the environmental quality at
state i. Then, the willingness to pay function is defined as a difference of two expenditure
14
Since the purpose of reporting willingness to pay is to compare the result from each estimation and decision
model, monetary units are ignored in the table. Furthermore, by the assumption of linear function and infinite
range of error distribution, the expected willingness to pay can be negative value.
12
function: WTP ( u 0 ) = m ( q 0 , u 0 ) − m ( q1 , u 0 ) . Like the random utility, the expenditure
function consists of a systematic component ( mn* ) and an unobservable random component
( ηn ). The logistic distribution of the willingness to pay function implies that the underlying
distribution of expenditure functions is the i.i.d. type I extreme value distribution. As can be
recognized, the exactly same problems as random utility model arise in the willingness to
pay function model.
While the random utility is derived from the utility maximization, the expenditure
function is the minimized cost, requiring that the extreme value of the expenditure function
is the smallest extreme value. The smallest extreme value is easily derived from the dual
relation of min ( Z i ) = − max ( − Z i ) (Tiago de Oliveira, 1983). The joint distribution
function for minima with Gumbel reduced margins is
Ω (η0 ,η1 ) = 1 − F ( −η0 ) − F ( −η1 ) + F ( −η0 , −η1 ) ,
where F ( ⋅) and F ( ⋅, ⋅) are marginal and joint distributions of maxima. The probability
density function of bivariate extreme values of minima is ω (η0 ,η1 ) = f ( −η0 , −η1 ) by
definition. From the dual relationship, we can derive the marginal distribution function and
probability function as Ω (ηi ) = 1 − F ( −ηi ) and ω (η ) = f ( −η ) . Note that the expected
value of η is E (ηi ) ≈ −0.57722θ i . The conditional distribution of minima is derived from
the conditional distribution of maxima in equation (9) such as
Ω (η1 | η0 ) = ∫
η =η1
η =−∞
ω (η | η0 ) dη = 1 − F ( −η1 | −η0 )
since ω (η1 | η0 ) = f ( −η1 | −η0 ) . The distribution and probability functions of reduced
difference of minima are identical to that of maxima.
The estimation of the willingness to pay function follows the same procedures in
random utility model. The choice probability of expenditure difference is expressed as
P1n = P ( m1*n + bn + η1 < m0*n + η0 ) = ∫
η0 =+∞
η0 =−∞
Ω ( mn* − bn + η0 | η0 ) ω (η0 ) dη0 .
where mn* = m0*n − m1n* . The choice probability can be approximated by Gaussian quadrature
L
such as Pˆ1n = ∑ l =1 ξl H ( mn* , ul ) through appropriate transformation. The mixed logit model
13
simulates the choice probability as P1n = ∫ L1n (ς ) ω (η ) dη where η = η0 ,η1 and
{
(
)
}
L1n = 1 + exp ⎡⎣ − mn* − bn / s − (η n 0 − η n1 ) / s ⎤⎦
−1
. The simulation procedure of the bivariate
probability of minima is the same as the bivariate probability of maxima. However, all
parameters are normalized by θ1 rather than by θ 0 such as β = ( β 0 − β1 ) / θ1 for the
systematic part, βb = 1/ θ1 for the minus bid value and γ = θ 0 / θ1 for the relative scale
factor. Due to the bid variable, the expenditure difference model is able to identify both
scale parameters. Finally, the expected willingness to pay is estimated as
E (WTPn ) = mn* + E (η0 − η1 ) .
Table 4 and Table 5 report the estimation results of expenditure difference model
using the same data in the random utility model. Note that the relative scale factor γ is
estimated as θ1 / θ 0 rather than θ 0 / θ1 to enable the comparison with random utility model.
Parameter estimates of the expenditure difference model are statistically duplicates of the
random utility model, i.e. assumption of underlying distribution such as maxima or minima
does not affect the estimation result. Table 4 shows that Barbados study satisfies the
classical assumption of logit model in terms of parameter estimates and LR test statistics.
Montevideo data in Table 5, however, rejects the homoskedasticity constraint but fails to
reject the independence constraint in both estimation models as the random utility model
(Table 2). However, in spite of that the relative scale estimate is statistically significantly
less than one, parameter estimates of systematic component of expenditure difference are
seemingly equivalent with that of the random utility model15. The expected willingness to
pay in expenditure difference model is reported in Table 3.
[Table 4-5 located here]
6. Conclusions
We challenge the theoretical and technical background of the simple logit model
often used for estimating willingness to pay from dichotomous choice contingent valuation
applications. The simple logit model assumes that the respondent’s evaluations of the two
15
Note that the parameters are normalized by θ1 not by θ0 .
14
states are stochastically independent and homoskedastic. However, when random utilities
across states of the world are heteroskedastic or correlated, we cannot derive the logit
model theoretically consistent with random utilities. In this paper, we suggest generalized
estimation methods by utilizing Gumbel mixed model to relax restrictive assumptions of
the standard random utility model. Nested within this generalized model are the
heteroskedastic logit model and the simple logit. The nesting structure allows for
straightforward tests of the homoskedastic-independent error assumptions. In addition to
the random utility model, expenditure difference (willingness to pay) model was estimated
using Gumbel mixed bivariate distribution of minima. Again, this model has nested within
it a number of standard logit-expenditure difference models.
Estimation results from several existing data including Barbados and Montevideo
data show that correlation between two states is usually minimal, but Montevideo data
presents extremely different scale of error terms across states implying that the i.i.d.
extreme values, i.e. logistic distribution for the difference of error terms, may not be a
suitable distribution. Serious problem arises in estimation of welfare measure.
Heteroskedasticity or correlation provides willingness to pay estimate different from
estimate of the simple logit, thus different policy implication in benefit-cost analysis.
In spite of the simplicity and profound theory of binary choice logit model, much
careful consideration is required to apply the model into contingent valuation studies.
Various estimation models do not suggest different decision process but indicate that due to
the nature of decision process, the estimation result from simple logit model could be
incorrect. However, as we mentioned before, if random utility or expenditure function
follows the bivariate normal distribution, the binary choice probability is still a normal
distribution. The similar estimation result between probit and logit models arises the
question about the comparison between models with bivariate normal and bivariate extreme
value distributions. Decision of which estimation model should be used in practice is based
on the how researchers define the choice situation and choice set, but when they employ the
logit model, the assumption of the model should be tested in priori.
15
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19
Figure 1: Distribution Function of Gumbel Mixed Model of Maxima with λ = 0.5
2.5
0.1
0.5
0.4
2
0.8
0.6
0.3
0.7
1.5
0.2
1
0.6
0.5
0.5
0.4
0.1
ε1
0.3
0
0.3
0.2
0.2
-0.5
0.1
0.1
-1
-1.5
-2
-2.5
-2.5
-2
-1.5
-1
-0.5
0
ε0
0.5
1
1.5
2
2.5
Figure 2: Probability Function of Gumbel Mixed Model of Maxima with λ = 0.5
2.5
0.02
0.02
0. 0
2
2
0.04
0.
1.5
0.06
0.0
0 02
0.14
0.06
4
06
0. 14
0.
ε1
2
0 .1
0 .0 8
0.12
0.1
0 .0
0 .0 2
-0.5
8
0
0. 1
0.0
0.12
0. 1
0.5
4
0.08
-1
0.06
-1.5
0.0
1
04
6
0.08
4
0.0
2
0. 0
0.02
0. 04
0.02
-2
-2.5
-2.5
-2
-1.5
-1
-0.5
0
ε0
20
0.5
1
1.5
2
2.5
Figure 3: Distribution Function of Reduced Difference of Extreme Values
1
0.9
λ=1→
0.8
0.7
0.6
0.5
0.4
0.3
λ=0→
0.2
0.1
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
τ = ε0/ θ0 - ε1/θ1
Figure 4: Probability Function of Reduced Difference of Extreme Values
0.45
λ=1→
0.4
0.35
0.3
0.25
↑
λ=0
0.2
0.15
0.1
0.05
-2
-1.5
-1
-0.5
0
τ = ε0/ θ0 - ε1/θ1
21
0.5
1
1.5
2
Table 1. Estimation Result of Random Utility Model with Barbados Data
(γ = 1)
Bivariate
Extreme
Value Model
(γ = 1, λ = 0)
0.6790
0.5262
0.0549
0.0210
0.4100
0.2929
-0.0285
0.0090
0.0387
0.0063
1.0000
0.0000
-
0.5959
0.4233
0.0412
0.0178
0.2804
0.2304
-0.0219
0.0083
0.0311
0.0082
1.0000
0.6099
0.3739
0.6700
0.5861
0.0498
0.0375
0.3401
0.3602
-0.0272
0.0209
0.0389
0.0283
1.2648
0.8285
0.5638
0.4613
0.4718
0.5207
0.0645
0.0238
0.4527
0.3150
-0.0265
0.0098
0.0408
0.0078
1.0000
0.0000
-
0.5803
0.4771
0.0532
0.0233
0.3432
0.3137
-0.0250
0.0092
0.0368
0.0068
1.0000
0.3516
0.5122
0.6646
0.7226
0.0812
0.0703
0.5297
0.6748
-0.0386
0.0232
0.0554
0.0355
1.6949
0.9574
0.2623
0.6528
-160.841
-160.211
-160.122
-160.782
-160.678
-160.268
Constrained Bivariate
Logit
(γ = 1, λ = 0)
Constant
Income
City
Age
Bid
0.6789
0.5261
0.0549
0.0210
0.4099
0.2929
-0.0285
0.0090
0.0387
0.0063
γ
-
λ
-
Log likelihood
*
-160.841
( λ = 0)
*
Constrained Mixed Logit
( λ = 0)
*
(γ = 1)
Mixed Logit
Model
Function calculation is failed.
Table 2. Estimation Result of Random Utility Model with Montevideo Data
(γ = 1, λ = 0)
( λ = 0)
(γ = 1)
Bivariate
Extreme
Value Model
-0.6139
0.1295
0.0944
0.0192
0.0100
0.0014
1.0000
0.0000
-
-0.1297
0.0822
0.0743
0.0138
0.0053
0.0008
0.0000
.
0.0000
-
-0.4860
0.1444
0.0778
0.0228
0.0078
0.0022
1.0000
0.5313
0.4445
-0.1297
0.0823
0.0743
0.0138
0.0053
0.0008
0.0000
.
0.1909
0.0000
-0.6985
0.1336
0.1108
0.0183
0.0104
0.0015
1.0000
0.0000
-
-0.1360
0.0924
0.0812
0.0146
0.0063
0.0009
0.0000
.
0.0000
-
-0.6565
0.1617
0.1074
0.0286
0.0096
0.0017
1.0000
0.2277
0.2478
-0.0994
0.0819
0.0708
0.0077
0.0063
0.0009
0.0000
.
1.0000
.
-726.720
-723.611
-726.244
-723.611
-726.174
-723.625
-725.782
-723.432
Constrained Bivariate
Logit
Constant
Income
Bid
-0.6140
0.1295
0.0944
0.0192
0.0100
0.0014
γ
-
λ
-
Log likelihood
-726.719
22
Constrained Mixed Logit
(γ = 1, λ = 0)
( λ = 0)
(γ = 1)
Mixed Logit
Model
Table 3. Sample Average of Welfare Measure for Environmental Quality Change
(γ = 1, λ = 0)
( λ = 0)
(γ = 1)
Bivariate
Extreme
Value Model
Constrained Bivariate
Logit
Constrained Mixed Logit
(γ = 1, λ = 0)
( λ = 0)
(γ = 1)
Mixed Logit
Model
Barbados Data
RU1
-2.0701
-2.0688
.
-0.2368
1.6110
-2.1668
.
-1.1496
1.6930
ED2
-2.0701
-2.0709
7467.3236
-0.2374
2.8939
-3.4701
3.7356
-0.0794
2.9927
-26.8148
-26.8149
-81.6514
-25.7220
-81.6283
-27.9171
-65.7042
-27.2881
-65.1584
-26.8148
-26.8189
.
-25.7257
.
-27.3092
-65.3239
-26.4295
-65.3239
Montevideo Data
RU1
2
ED
1
2
RU represents the random utility model.
ED represents the expenditure difference model.
Table 4. Estimation Result of Expenditure Difference Model with Barbados Data
(γ = 1, λ = 0)
( λ = 0)
(γ = 1)
Bivariate
Extreme
Value Model
0.6789
0.5260
0.0549
0.0210
0.4099
0.2929
-0.0285
0.0090
0.0387
0.0063
1.0000
0.0000
-
0.3512
0.4404
0.0419
0.0155
0.3047
0.2342
-0.0244
0.0075
0.0351
0.0055
455.6437
42029.933
0.0000
-
0.5958
0.4233
0.0412
0.0178
0.2804
0.2304
-0.0219
0.0083
0.0311
0.0082
1.0000
0.6099
0.3740
0.5283
0.4392
0.0393
0.0173
0.2687
0.2228
-0.0215
0.0082
0.0307
0.0083
1.2726
0.8559
0.5630
0.4635
0.5658
0.5402
0.0569
0.0216
0.4682
0.2981
-0.0281
0.0088
0.0385
0.0046
1.0000
0.0000
-
1.1122
1.1487
0.1063
0.0513
0.9152
0.6835
-0.0594
0.0346
0.0845
0.0390
2.4492
1.1828
0.0000
-
0.6474
0.4580
0.0494
0.0185
0.3478
0.2690
-0.0249
0.0081
0.0366
0.0053
1.0000
0.4651
0.3311
0.7489
0.7095
0.0649
0.0396
0.4112
0.4621
-0.0330
0.0235
0.0510
0.0297
1.4920
0.8958
0.4246
0.3744
-160.841
-160.171
-160.211
-160.121
-160.320
-159.267
-159.667
-158.996
Constrained Bivariate
Logit
Constant
Income
City
Age
Bid
0.6789
0.5261
0.0549
0.0210
0.4099
0.2929
-0.0285
0.0090
0.0387
0.0063
γ
-
λ
-
Log likelihood
-160.841
23
Constrained Mixed Logit
(γ = 1, λ = 0)
( λ = 0)
(γ = 1)
Mixed Logit
Model
Table 5. Estimation Result of Expenditure Difference Model with Montevideo Data
Bivariate
Extreme
Value
Model*
Constrained Bivariate
Logit
(γ = 1, λ = 0)
Constant
Income
Bid
-0.6140
0.1295
0.0944
0.0192
0.0100
0.0014
γ
-
λ
-
Log likelihood
*
-726.719
( λ = 0)*
(γ = 1)
Constrained Mixed Logit
(γ = 1, λ = 0)
( λ = 0)
(γ = 1)
Mixed Logit
Model
-0.6140
0.1295
0.0944
0.0192
0.0100
0.0014
1.0000
0.0000
-
-0.4860
0.1444
0.0778
0.0228
0.0078
0.0022
1.0000
0.5313
0.4445
-0.6757
0.1392
0.1064
0.0214
0.0104
0.0015
1.0000
0.0000
-
-0.1988
0.1462
0.0824
0.0153
0.0064
0.0009
0.1009
0.1913
0.0000
-
-0.6238
0.1312
0.1110
0.0222
0.0082
0.0016
1.0000
0.6041
0.3006
-0.1988
0.1462
0.0824
0.0153
0.0064
0.0009
0.1009
0.1911
0.0000
.
-726.719
-726.243
-726.269
-723.976
-724.497
-723.976
Function calculation is failed.
24